arcsiny

arcsiny is the inverse sine function. It assigns to a real number y in the interval [-1, 1] the unique angle x in [-π/2, π/2] whose sine equals y. Consequently, arcsiny maps [-1, 1] to [-π/2, π/2] and is defined only for y within that interval. It is the inverse of the sine function when the latter is restricted to the interval [-π/2, π/2], where sine is strictly increasing and thus invertible. The function is typically denoted arcsin(y) or sin^(-1)(y); some programming languages implement it as asin(y). The output is usually given in radians unless otherwise stated.

Key properties: arcsiny is an odd function: arcsiny(-y) = -arcsiny(y). It satisfies sin(arcsiny(y)) = y for all y in [-1, 1], and arcsiny(sin x) = x for all x in [-π/2, π/2]. The derivative is d/dy arcsiny(y) = 1 / sqrt(1 - y^2) for |y| < 1, with vertical tangents at y = ±1. The endpoint values are arcsiny(-1) = -π/2, arcsiny(0) = 0, arcsiny(1) = π/2.

Numerical evaluation: arcsiny has a convergent power series around 0: arcsiny(y) = y + y^3/6 + 3y^5/40 + 5y^7/112 + … It can also be computed by standard algorithms using range reduction and polynomial or rational approximations.

Applications: arcsiny is used to determine angles from a known sine value, in geometry, trigonometric computations in graphics, physics, and engineering.